How to use the z score calculator
- 1 Enter the raw score (x).
- 2 Enter the mean (μ) of the distribution.
- 3 Enter the standard deviation (σ).
- 4 Read the z-score, the percentile and the p-values.
Worked example
A test score of 85 where the mean is 70 and the standard deviation is 10. z = (85 − 70) ÷ 10 = 1.5, so the score is 1.5 standard deviations above the mean, at the 93.32nd percentile. The area to the right is 6.68%, and the two-tailed p-value is 0.1336.
Reading a z-score
- Positive is above the mean, negative below.
- About 95% of values fall between −2 and +2.
- Beyond ±3 is very unusual.
- Use the sample standard deviation for sample data.
Get the mean and spread with the Mean, Median and Mode Calculator, build an interval with the Confidence Interval Calculator, and combine chances with the Probability Calculator. Medical scores are often indexed to body size: see the Body Surface Area Calculator.
Features
No z-table needed
Normal distribution areas are calculated for you.
Percentile and p-values
Left area, right area, one- and two-tailed p-values.
Plain explanation
How many standard deviations above or below the mean.
When to use it
Comparing test scores
See which score is better on different scales. Get the mean and spread with the Mean, Median and Mode Calculator.
Statistics homework
Check z-tests and normal distribution problems. Build intervals with the Confidence Interval Calculator.
Spotting outliers
Values beyond ±3 standard deviations are unusual.
Standard deviation first
Work it out with the Standard Deviation Calculator.
Related Math tasks
Frequently asked questions
What is a z-score?
A measure of how many standard deviations a value is above or below the mean. A z-score of 1.5 means 1.5 standard deviations above the average.
How do you calculate a z-score?
Subtract the mean from the value and divide by the standard deviation: z = (x − μ) ÷ σ. For x = 85, μ = 70 and σ = 10, z = 1.5.
What percentile is a z-score of 1.5?
About the 93rd percentile: 93.32% of a normal distribution lies below z = 1.5.
What does a negative z-score mean?
The value is below the mean. z = −1 is one standard deviation below average, around the 16th percentile.
What is a good z-score?
It depends on the context. For test scores, a positive z-score is above average; z = 2 is better than about 97.7% of scores.
How do I get a p-value from a z-score?
The one-tailed p-value is the area beyond z on one side; the two-tailed p-value doubles it. For z = 1.5, they are 0.0668 and 0.1336.
What z-score is significant?
In a two-tailed test at the 5% level, |z| greater than 1.96 is significant; at the 1% level, greater than 2.576.
What is the empirical rule?
In a normal distribution, about 68% of values lie within 1 standard deviation of the mean, 95% within 2 and 99.7% within 3.
Should I use the population or sample standard deviation?
Use the population standard deviation if you know it. With a small sample and an estimated standard deviation, a t-score is more accurate.
Does the data have to be normally distributed?
The z-score itself works for any data, but the percentile and p-values assume a normal distribution.
What is the z-score for the 95th percentile?
About 1.645. The 97.5th percentile is 1.96, which is why 1.96 is used for 95% confidence intervals.
Can a z-score be greater than 3?
Yes, but it is rare in a normal distribution: only about 0.13% of values lie above z = 3.
How do I compare two test scores?
Convert each to a z-score with its own mean and standard deviation. The higher z-score is the better relative result.
Built and reviewed by Hamza Farooq, founder and lead developer
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